45 Electromagnetic Energy

The energy stored in electric and magnetic fields that is used to account for work done by electromagnetic forces and for energy carried by waves.

The electromagnetic energy density. The energy stored in electromagnetic fields has a local density built from \(E^{2}\) and \(B^{2}\). This principle is used to compute the field energy in a region by integration.

The electromagnetic energy density is

\[ u = \dfrac{1}{2}\left(\epsilon_{0}E^{2} + \dfrac{1}{\mu_{0}}B^{2}\right) \]

where

\(u\) is the energy density.
\(E\) is the magnitude of the electric field.
\(B\) is the magnitude of the magnetic field.
\(\epsilon_{0}\) is the permittivity of free space.
\(\mu_{0}\) is the permeability of free space.

The total field energy. The total field energy is the integral of that density over space. This principle is used to assign a single energy to a field configuration.

The total field energy is

\[ U = \displaystyle\int u\,d\tau \]

where

\(U\) is the total electromagnetic energy.
\(u\) is the energy density.
\(d\tau\) is the volume element.

Poynting’s theorem. Energy leaves a volume as the flux of the Poynting vector. This principle is used to write local conservation of electromagnetic energy.

Poynting’s theorem is

\[ \dfrac{\partial u}{\partial t} + \nabla\cdot\mathbf{S} = -\mathbf{J}\cdot\mathbf{E} \]

where

\(u\) is the energy density.
\(\mathbf{S}\) is the Poynting vector.
\(\mathbf{J}\) is the current density.
\(\mathbf{E}\) is the electric field.
\(t\) is time.

Note: Also called field energy.

45.1 References

Griffiths, D. J. Introduction to Electrodynamics. Cambridge University Press, 2024. — \(u=\dfrac{1}{2}(\epsilon_{0}E^{2}+B^{2}/\mu_{0})\).
Knight, R. D. Physics for Scientists and Engineers. Pearson, 2023. — electric and magnetic energy densities.
Susskind, L., & Friedman, A. Special Relativity and Classical Field Theory. Basic Books, 2017. — energy in the electromagnetic field.

44 Electric Potential

46 Electromagnetic Induction